How to subtract smaller face from a bigger face in edit mode?
If you have a solid face and you want to poke a hole on it and conform it to another existing shape (target), you can inset (I in edit mode) the face to be poked, delete (x) the resulting face and then snap the remaining vertices to the target face one by one.
The problem you present, though, requires a different strategy. Instead of selecting all vertices and hitting F to make faces out of them, select vertices by smaller groups that make sense and are more or less free of concavities. First the ones which inset into the desired empty space (the ones that make the wheels' fork end). Select just them and hit F. Then to for small sections of the wheel until you cover what you want covered.
Pay attention so you don't select vertices that shouldn't be adjacent to the new face you're about to create. Blender lends itself to easily create bad topology if the users direct it to do so.
Later, you can fuse the faces together as you wish by dissolving shared edges between them (X).